Finally, we analyze the effect of the atomic relaxation induced pseudo-vector potential terms on the narrow bandwidth by studying the first order model. Such terms appear already at the first order in gradient expansion, so there is no justification for dropping them in the BM model with the relaxation. Because the pseudo-vector potential terms are of the same order as the contact inter-layer tunneling terms \(w_{0,1}\) , one may naively conclude that their effect is to broaden the bandwidth by a similar order and to prevent the magic angle phenomenon. While they do increase the bandwidth at the “old” magic angle (i.e. without the periodic relaxation induced vector potential), we find that their effect can be compensated by a change of the twist angle, recovering the narrow band at a new (smaller) magic angle. We were able to demonstrate this by solving the problem analytically in the chiral limit including the pseudo-vector potential terms absent in Ref. [1]}. We highlight the importance of \(C_3\)  symmetry for this compensation.
